Compact derivations relative to semifinite von Neumann algebras. (English) Zbl 0578.46055
B. Johnson and S. Parrott [J. Funct. Anal. 11, 39-61 (1972; Zbl 0237.46070)] have proven that if \(A\subset B(H)\) is a von Neumann algebra that does not contain as a direct summand certain type \(II_ 1\) factors and if \(\delta\) is a derivation from A into the compact operators of B(H), then \(\delta\) is implemented by a compact operator; i.e., \(\delta\) is inner.
S. Popa [The commutant modulo the set of compact operators of a von Neumann algebra, Preprint] has recently extended the result to all von Neumann algebras A. In the paper under review, it is proven that if B is a von Neumann algebra, A is an abelian or properly infinite subalgebra of B containing the center of B and \(\delta\) is a derivation from A into either the compact ideal J(B) of B (the ideal generated by the finite projections of B) or into the Cp(B,\(\tau)\)-Schatten class of B relative to a trace \(\tau\), then \(\delta\) is inner.
In addition, the condition that A contains the center of B can be removed for the Cp(B,\(\tau)\) case or the case when A is an abelian atomic algebra.
S. Popa [The commutant modulo the set of compact operators of a von Neumann algebra, Preprint] has recently extended the result to all von Neumann algebras A. In the paper under review, it is proven that if B is a von Neumann algebra, A is an abelian or properly infinite subalgebra of B containing the center of B and \(\delta\) is a derivation from A into either the compact ideal J(B) of B (the ideal generated by the finite projections of B) or into the Cp(B,\(\tau)\)-Schatten class of B relative to a trace \(\tau\), then \(\delta\) is inner.
In addition, the condition that A contains the center of B can be removed for the Cp(B,\(\tau)\) case or the case when A is an abelian atomic algebra.
MSC:
| 46L10 | General theory of von Neumann algebras |
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 47C15 | Linear operators in \(C^*\)- or von Neumann algebras |
Citations:
Zbl 0237.46070References:
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